Theorem sseqtrrd 3209

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2195	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3208	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  sseqtrrid  3221  fnfvima  5772  tfrlemiubacc  6356  tfr1onlemubacc  6372  tfrcllemubacc  6385  rdgivallem  6407  nnnninf  7155  nninfwlpoimlemg  7204  dfphi2  12255  ctinf  12484  imasaddfnlemg  12794  imasaddvallemg  12795  subsubm  12950  subsubg  13153  subsubrng  13578  subsubrg  13609  lidlss  13809  toponss  14003  ssntr  14099  iscnp3  14180  cnprcl2k  14183  tgcn  14185  tgcnp  14186  ssidcn  14187  cncnp  14207  txcnp  14248  imasnopn  14276  hmeontr  14290  blssec  14415  blssopn  14462  xmettx  14487  metcnp  14489  nnsf  15233  nninfsellemsuc  15240